prove that - the radii of incircles of two similar triangles are proportional to the corresponding sides
Answers
the radii of incircles of two similar triangles are proportional to the corresponding sides
Step-by-step explanation:
Let say ΔABC ≈ Δ PQR
r₁ - radius of incircle of ΔABC
r₂ - radius of incircle of ΔPQR
AB/PQ = BC/QR = AC/PR = k
=> Area of ΔABC / Area of ΔPQR = k²
Area of ΔABC = (1/2)(AB + BC + AC) * r₁
Area of ΔPQR = (1/2)(PQ +QR+ PR) * r₂
(1/2)(AB + BC + AC) * r₁ / (1/2)(PQ +QR+ PR) * r₂ = k²
=> (AB + BC + AC) * r₁ / (PQ +QR+ PR) * r₂ = k²
AB = kPQ
BC = kQR
AC = kPR
=> (kPQ +kQR+ kPR) * r₁ / (PQ +QR+ PR) * r₂ = k²
=> k(PQ +QR+ PR)* r₁ / (PQ +QR+ PR) * r₂ = k²
=> r₁ / r₂ = k
=> AB/PQ = BC/QR = AC/PR = r₁ / r₂
=> the radii of incircles of two similar triangles are proportional to the corresponding sides
QED
Proved
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